DOCSLIB.ORG

Quantum Information & Quantum Computation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quantum Information & Quantum Computation CS290A, Spring 2005: Quantum Information & Quantum Computation Wim van Dam Engineering 1, Room 5109 vandam@cs http://www.cs.ucsb.edu/~vandam/teaching/CS290/ Administrative • The Final Examination will be: Monday June 6, 12:00–15:00, PHELPS 1401 • New Exercises are posted Try to answer Question 2 before Thursday. Last Week / This Week • Last week we looked at quantum money and quantum cryptography, which uses the qubit states 0,1,+,–. • This week we will extend this idea to describe “quantum fingerprinting”. • Also this week: superdense quantum coding and quantum teleportation of quantum states. Fingerprinting Assume two parties A and B that each have data in the form of a (long) string x and y ∈{0,1} N. A and B want to check if they have the same data, without revealing a priori to the other their strings. They do this by sending (publicly) information about their strings (x and y) to a trusted third party C, who decides. Sending the whole strings is not allowed because the strings are too long / risk of eavesdropping. A and B want to have a way of fingerprinting their strings. Quantum Fingerprinting “Quantum Fingerprinting” refers to a way of mapping the ∈ N strings x,y {0,1} to quantum states | φx〉 and | φy〉, that live in a ‘much smaller than 2 N’-dimensional Hilbert space, such that from ψ and φ we can tell decide whether x=y. ∈ N The set { |φx〉 : x {0,1} } cannot be mutually orthogonal. Instead we will have to work with near orthogonal states. ∈ N Central Idea: Encode x {0,1} into m qubit state |φx〉 (with m much smaller than N). 2 Do this in a way such that | 〈φx|φy〉| ≤ ½ if x≠y. Third party decides if |φx〉 = |φy〉 or not. Simple Example Let x and y be from a set of 6 possibilities. Let | φx〉 and |φy〉 be qubits from the set of 6 states {|0〉, |1〉, (|0〉+|1〉)/ √2, (|0〉–|1〉)/ √2, (|0〉+i|1〉)/ √2, (|0〉–i|1〉)/ √2} 2 For all qubit states with x≠y we have |〈φx|φy〉| ≤ ½. The third party receives two unknown states |φx〉 and |φy〉 that are either the same or very different. How to distinguish between these two possibilities? A Quantum State Equality Tester for unknown states can be implemented with a Controlled Swap Test… Controlled Swap Test • Given two unknown quantum states | ψ〉 and | φ〉, are they the same or not? • You can test this using the Controlled Swap Test” C-SWAP:|0,x,y 〉 # |0,x,y 〉 |0 |0 ? C-SWAP:|1,x,y 〉 # |1,y,x 〉 〉 H H 〉 in the circuit……………. | S φ〉 W | A ψ〉 P • Observing a “0” indicates that |ψ〉 and | φ〉 are close to each other, observing a “1” that they are far apart. • What are the exact probabilities? Probabilities of C-SWAP • The evolution of the system is |0 〉 H H |0 〉? ,0 , a 1 ( ,0 , + ,1 , ) | S φ ψ 2 φ ψ φ ψ φ〉 W A 1 | 〉 a ( ,0 , + ,1 , ) ψ P 2 φ ψ ψ φ a 1 + + − 2 ( ,0 φ,ψ ,1φ,ψ ,0 ψ,φ ,1ψ,φ ) = ⊗ 1 + + ⊗ 1 − 0 2 ( φ,ψ ψ,φ ) 1 2 ( φ,ψ ψ,φ ) The probability of observing a “0” is therefore = 1 + + Prob("0" ) 4 ( φ,ψ ψ,φ )( φ,ψ ψ,φ ) = 1 + + 4 2( ψ,φ φ,ψ φ,ψ ψ,φ ) 2 = 1 + 1 2 2 ψ φ Equality Testing • The previous calculations |0 〉 H H |0 〉? show that if | , |2 1, 〈φ ψ〉 ≈ |φ〉 S then the probability of W |ψ〉 A observing a “0” is ≈ 1. P • If | 〈φ,ψ〉|2 ≈ 0, then the probability of “0” is ≈ ½. • By repeating the experiment a number of times (using fresh copies of | ψ〉 and | φ〉), we can –with near certainty– distinguish between the cases |ψ〉 = | φ〉 and |〈φ,ψ〉|2 ≤ ½. More on Q-Fingerprinting Using methods from error correction it is possible to encode N bits of information into M ≈ c log N qubits 2 such that |〈φx|φy〉| ≤ ½ if x≠y. Even if we have to send many copies of the the fingerprint, it will still be more efficient than sending the N classical bits of the original strings x and y. Added advantage: because we send such a highly compressed quantum state, it is impossible to infer the string x from the fingerprint |φx〉. Communication & Entanglement • Holevo’s bound tells us that we can encode only one bit of information into one qubit. This bound assumes that the qubit is unentangled. • Things change if we allow the communication of qubits that are entangled with qubits of the receiver. • What happens if A and B share a prior entangled pair of qubits |EPR 〉 = (|0 A,0 B〉+|1 A,1 B〉)/ √2 before Alice is to send (quantum) information to Bob? Superdense Quantum Coding • Let A and B share an EPR pair of qubits. Alice wants to transmit classical information to Bob. • Using Superdense coding, A can send two bits of information to B, using only one qubit. • Approach: Depending on A’s input {1,2,3,4} she applies to her side of the EPR pair, one of 4 transformations { I, X, Y, Z}, and sends her EPR qubit to Bob. • From the changed EPR pair, Bob decodes which transformation A has applied. This can be done reliably and will transfer 2 bits of information from A to B. How to Do It? • Look back at Question 1, Exercises 2 (Week 3) )I( a 1 ( 00 + 11 ) • The four transformations 2 give us four entangled (X) a 1 ( 10 + 01 ) states that are mutually 2 orthogonal… (Y) a 1 10i( − i 01 ) 2 )Z( a 1 ( 00 − 11 ) 2 • With a CNOT and a Hadamard, Bob can map those states to the outputs |00 〉, |01 〉, –i|11 〉 |and |10 〉. More on Superdense Coding • One can prove that the 2 bits / 1 qubit ratio is optimal: it is not possible to send (say) 3 bits of information using 1 qubit and lots of entanglement. • Superdense coding is not possible classically. • Experimental implementation of superdense coding: [Zeilinger et al.,1996, Innsbruck, Austria] • Can we do the inverse: Send quantum information using classical communication between A and B? What is Teleportation? Alice unknown q qubit = α 0 + β 1 Alice wants to send her unknown quantum information to Bob A and B do not have a quantum channel: only classical communication is allowed. Alice cannot tell Bob what the values nor can she measure |q Bob. 〉 to see what they are. Solution: Use α,β are, an entangled EPR pair Towards Teleportation The first two qubits are on Alice’s side The 3 rd one is Bob’s After H/CNOT, they share an EPR pair. Alice performs a CNOT and a Hadamard to her 2 qubits and measures them in the 0/1 basis. What can she expect to observe? ?tuOtup q ⊗ EPR = ( 0 + 1 ) ⊗ 1 ( 00 + 11 ) α β 2 = 1 0( , 00+0 , 11+1 , 00+1 , 11) 2 α α β β a 1 0( , 00+0 , 11+1 , 01+1 , 10) 2 α α β β a 1 + + + + 20(α , 001α , 000α , 111α , 11 0β , 01−1β , 01+0β , 10−1β , 10) adeerRlgss of = 1 ⊗ + + 2 00 (α 0 β 1 ) thauele s vα,β 1 10 ⊗ ( 1 + 0 ) + tpho reb atibyli 2 α β fmonuieg ras 1 ⊗ − + 2 01 (α 0 β 1 ) 0{0,,101110}, 1 ⊗ − s iounaetqer.- r 2 11 (α 1 β 0 ) Effect of A’s Measurement = 1 00 ⊗ ( 0 + 1 ) + 2 α β Depending on the two 1 ⊗ + + leftmost qubits (A’s side), 2 01 (α 1 β 0 ) the third qubit (B’s side) 1 ⊗ − + 2 10 (α 0 β 1 ) is 1-out of-4 permutations 1 ⊗ − of the original qubit |q . 2 11 (α 1 β 0 ) 〉 When A measures the Outcome “00”: → α|0 〉+β|1 〉 two bits ∈{00,01,10,11} Outcome “01”: → α|1 〉+β|0 〉 the qubit of B collapses Outcome “10”: → α|0 〉–β|1 〉 to 1 of the 4 versions of q. Outcome “11”: → α|1 〉–β|0 〉 When A tells B which one of the 4 outcomes she has observed, Bob knows what to do to correct his qubit to the original |q 〉. Bob’s Correction “00”: → α|0 〉+β|1 〉 When A tells B which one of the “01”: → α|1 〉+β|0 〉 4 outcomes she has observed, “10”: → α|0 〉–β|1 〉 Bob knows what to do to correct “11”: → α|1 〉–β|0 〉 his qubit to the original |q 〉… If first bit is a “1”, Bob applies a Z gate |b 〉 # (–1) b|b 〉. If 2nd bit is a “1”, Bob applies a NOT |b 〉 # |b ⊕1〉 The outcome will always be α|0 〉+β|1 〉 = |q 〉 Bob has (re)created the unknown qubit q that was on Alice’s side. During the process Alice has ‘lost’ her copy of the qubit (otherwise we would have copied q). Teleportation Alice Shared EPR pair Bob unknown qubit |q 〉 = α|0 〉+β|1 〉. G Alice applies a CNOT and a Hadamard to Bob receives the two q and her EPR half. bits and acts on his side of the EPR, in She measures the bits in a way that recreates the 0/1 basis and sends the unknown qubit the information to Bob. |q 〉 = α|0 〉+β|1 〉. What Just Happened? • Teleportation requires one EPR pair and two classical bits to transfer one qubit from A to B. • Superdense coding requires one EPR pair and one quantum bit to transfer two classical bits of information.
Load more

Recommended publications
  • Arxiv:1803.04114V2 [Quant-Ph] 16 Nov 2018
    Learning the quantum algorithm for state overlap Lukasz Cincio,1 Yiğit Subaşı,1 Andrew T. Sornborger,2 and Patrick J. Coles1 1Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. 2Information Sciences, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. Short-depth algorithms are crucial for reducing computational error on near-term quantum com- puters, for which decoherence and gate infidelity remain important issues. Here we present a machine-learning approach for discovering such algorithms. We apply our method to a ubiqui- tous primitive: computing the overlap Tr(ρσ) between two quantum states ρ and σ. The standard algorithm for this task, known as the Swap Test, is used in many applications such as quantum support vector machines, and, when specialized to ρ = σ, quantifies the Renyi entanglement. Here, we find algorithms that have shorter depths than the Swap Test, including one that has a constant depth (independent of problem size). Furthermore, we apply our approach to the hardware-specific connectivity and gate sets used by Rigetti’s and IBM’s quantum computers and demonstrate that the shorter algorithms that we derive significantly reduce the error - compared to the Swap Test - on these computers. I. INTRODUCTION hyperparameters, we optimize the algorithm in a task- oriented manner, i.e., by minimizing a cost function that quantifies the discrepancy between the algorithm’s out- Quantum supremacy [1] may be coming soon [2]. put and the desired output. The task is defined by a While it is an exciting time for quantum computing, de- training data set that exemplifies the desired computa- coherence and gate fidelity continue to be important is- tion.
    [Show full text]
  • Signing Perfect Currency Bonds
    Signing Perfect Currency Bonds Subhayan Roy Moulick∗ and Prasanta K. Panigrahiy Indian Institute of Science Education and Research Kolkata, Mohanpur 741246, West Bengal, India (Dated: May 23, 2019) We propose the idea of a Quantum Cheque Scheme, a cryptographic protocol in which any legiti- mate client of a trusted bank can issue a cheque, that cannot be counterfeited or altered in anyway, and can be verified by a bank or any of its branches. We formally define a Quantum Cheque and present the first Unconditionally Secure Quantum Cheque Scheme and show it to be secure against any no-signaling adversary. The proposed Quantum Cheque Scheme can been perceived as the quantum analog of Electronic Data Interchange, as an alternate for current e-Payment Gateways. PACS numbers: 03.67.Dd, 03.67.Hk, 03.67.Ac I. INTRODUCTION in addition to security against counterfeiters, owing its origin to the pioneering works of Mosca and Stebila [11]. Replication of classical information is a significant nui- They proposed a scheme based on blind quantum com- sance in copy-protection. Any physical entity created putation that required a verifier to do an obfuscated ver- classically can be, in principle, copied. Currency bonds, ification with the bank and learn only the validity of the printed on textile and paper, are no exception, and any quantum coin. This however is a private key protocol adversary, given sufficient time and resources, can be and requires communication with the bank. able to counterfeit currency bonds. However, the quan- In this paper we propose the idea of Quantum Cheques tum regime can circumvent this problem, exploiting the and present a construction of an Quantum Cheque `No Cloning Theorem' [1], and pave way for unforgeable Scheme with Perfect Security against any No-Signaling Quantum Currency that are impossible to counterfeit adversary.
    [Show full text]
  • Lecture 25: QMA(2) 1 Introduction 2 QMA(2): the 2-Prover
    Quantum Computation (CMU 18-859BB, Fall 2015) Lecture 25: QMA(2) December 7, 2015 Lecturer: Ryan O'Donnell Scribe: Yongshan Ding 1 Introduction So far, we have seen two major quantum complexity classes, namely BQP and QMA. They are the quantum analogue of two famous classical complexity classes, P (or more precisely BPP) and NP, respectively. In today's lecture, we shall extend to a generalization of QMA that only arises in the Quantum setting. In particular, we denote QMA(k) for the case that quantum verifier uses k quantum certificates. Before we study the new complexity class, recall the following \swap test" circuit from homework 3: qudit SWAP qudit qubit j0i H • H Accept if j0i When we measure the last register, we \accept" if the outcome is j0i and \reject" if the outcome is j1i. We have shown that this is \similarity test". In particular we have the following: 1 1 2 1 2 • Pr[Accept j i ⊗ j'i] = 2 + 2 j h j'i j = 1 − 2 dtr(j i ; j'i) 1 1 Pd 2 • Pr[Accept ρ ⊗ ρ] = 2 + 2 i=1 pi , if ρ = fpi j iig. 2 QMA(2): The 2-Prover QMA Recall from last time, we introduced the QMA class where there is a prover who is trying to prove some instance x is in the language L, regardless of whether it is true or not. Figure. 1 is a simple picture that describes this. The complexity class we are going to look at today involves multiple provers. Kobayashi et al.
    [Show full text]
  • Practical Quantum Digital Signature
    Practical Quantum Digital Signature 1,2, 1,2, 1,2, Hua-Lei Yin, ∗ Yao Fu, † and Zeng-Bing Chen ‡ 1Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China 2The CAS Center for Excellence in QIQP and the Synergetic Innovation Center for QIQP, University of Science and Technology of China, Hefei, Anhui 230026, China (Dated: September 5, 2018) Guaranteeing non-repudiation, unforgeability as well as transferability of a signature is one of the most vital safeguards in today’s e-commerce era. Based on fundamental laws of quantum physics, quantum digital signature (QDS) aims to provide information-theoretic security for this crypto- graphic task. However, up to date, the previously proposed QDS protocols are impractical due to various challenging problems and most importantly, the requirement of authenticated (secure) quan- tum channels between participants. Here, we present the first quantum digital signature protocol that removes the assumption of authenticated quantum channels while remaining secure against the collective attacks. Besides, our QDS protocol can be practically implemented over more than 100 km under current mature technology as used in quantum key distribution. PACS numbers: 03.67.Dd, 03.67.Hk, 03.67.Ac I. INTRODUCTION The authenticated quantum channels are equivalent to the secure quantum channels that do not allow any eaves- Digital signatures aim to certify the provenance and dropping. Recall that while quantum key distribution identity of a message as well as the authenticity of a (QKD) [11, 12] requires authenticated classical channels, signature. They are widely applied in e-mails, financial it does not require authenticated (secure) quantum chan- transactions, electronic contracts, software distribution nels because of potential eavesdropping.
    [Show full text]
  • Quantum Computing: Lecture Notes
    Quantum Computing: Lecture Notes Ronald de Wolf Preface These lecture notes were formed in small chunks during my “Quantum computing” course at the University of Amsterdam, Feb-May 2011, and compiled into one text thereafter. Each chapter was covered in a lecture of 2 45 minutes, with an additional 45-minute lecture for exercises and × homework. The first half of the course (Chapters 1–7) covers quantum algorithms, the second half covers quantum complexity (Chapters 8–9), stuff involving Alice and Bob (Chapters 10–13), and error-correction (Chapter 14). A 15th lecture about physical implementations and general outlook was more sketchy, and I didn’t write lecture notes for it. These chapters may also be read as a general introduction to the area of quantum computation and information from the perspective of a theoretical computer scientist. While I made an effort to make the text self-contained and consistent, it may still be somewhat rough around the edges; I hope to continue polishing and adding to it. Comments & constructive criticism are very welcome, and can be sent to [email protected] Attribution and acknowledgements Most of the material in Chapters 1–6 comes from the first chapter of my PhD thesis [71], with a number of additions: the lower bound for Simon, the Fourier transform, the geometric explanation of Grover. Chapter 7 is newly written for these notes, inspired by Santha’s survey [62]. Chapters 8 and 9 are largely new as well. Section 3 of Chapter 8, and most of Chapter 10 are taken (with many changes) from my “quantum proofs” survey paper with Andy Drucker [28].
    [Show full text]
  • 4. Quantum Computing
    4. QUANTUM COMPUTING Jozef Gruska Faculty of Informatics Brno Czech Republic October 11, 2013 Quantum Computing 4 - Quantum Gates and Circuits - 2013 4. QUANTUM CIRCUITS Quantum circuits are the most easy to deal with model of quantum computations. Several simple quantum gates form elementary building blocks from which any quantum circuit can be built. Jozef Gruska October 11, 2013 1 Quantum Computing 4 - Quantum Gates and Circuits - 2013 PART I - MOTIVATION Jozef Gruska October 11, 2013 2 Quantum Computing 4 - Quantum Gates and Circuits - 2013 MOTTO I. Progress in science is often done by pessimists. Progress in technology is always done by optimists. Jozef Gruska October 11, 2013 3 Quantum Computing 4 - Quantum Gates and Circuits - 2013 MOTTO II. Progress in science is often done by pessimists. Progress in technology is always done by knowledgeable and experienced optimists. Jozef Gruska October 11, 2013 4 Quantum Computing 4 - Quantum Gates and Circuits - 2013 TWO STORIES TO REMEMBER • The proposal to build Collosus, the first electronic computer for cryptanalysis purposes, was during the 2WW rejected by a committee of prominent specialists as impossible to make, in spite of the fact that British cryptanalysts needed it badly to crack communication between Hitler and his generals. • Collosus was then built by an ingenious optimist, Tommy Flowers, within 10 months in a Post office laboratory, and worked from the beginning successfully to break Lorenz cipher, starting January 1944. • The key point was that Flowers realized that velvets were reliable provided they were never switched on and off. (Of course, nobody believed him.) • The idea that 30m long ENIAC with 19000 vacuum tubes could work looked also crazy, for scientists, but it worked.
    [Show full text]
  • The Power of Unentanglement
    The Power of Unentanglement Scott Aaronson∗ Salman Beigi Andrew Drucker Bill Fefferman MIT MIT MIT University of Chicago Peter Shor MIT Abstract The class QMA (k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Does QMA (k) = QMA (2) for k 2? Can QMA (k) protocols be amplified to exponentially small error? ≥ In this paper, we make progress on all of the above questions. We give a protocol by which a verifier can be convinced that a 3Sat formula of size m is • satisfiable, with constant soundness, given O (√m) unentangled quantum witnesses with O (log m) qubits each. Our protocol relies on the existence of very short PCPs. We show that assuming a weak version of thee Additivity Conjecture from quantum infor- • mation theory, any QMA (2) protocol can be amplified to exponentially small error, and QMA (k)= QMA (2) for all k 2. ≥ We prove the nonexistence of “perfect disentanglers” for simulating multiple Merlins with • one. 1 Introduction Quantum entanglement is often described as a complicated, hard-to-understand resource. But ironically, many questions in quantum computing are easiest to answer assuming unlimited entan- glement, and become much more difficult if entanglement is not allowed! One way to understand this is that Hilbert space—the space of all quantum states—has extremely useful linear-algebraic properties, and when we restrict to the set of separable states we lose many of those properties.
    [Show full text]
  • QMA(2) Workshop— Tutorial 1 Bill Fefferman (Quics) Agenda
    QMA(2) workshop— Tutorial 1 Bill Fefferman (QuICS) Agenda I. Basics II. Known results III. Open questions/Next tutorial overview I. Basics I.1 Classical Complexity Theory • P • Class of problems efficiently solved on classical computer • NP • Class of problems with efficiently verifiable solutions P • Characterized by 3SAT • Input: Ψ:{0,1}n→{0,1} NP • n-variable 3-CNF formula • E.g., (x1∨x2∨x3)∧(x1∨−x2∨x6)∧... • Problem: ∃x1,x2,...,xn so that Ψ(x)=1? • Could use a box solving 3SAT to solve any problem in NP 4 I.2 Merlin-Arthur �∈{0,1}p(n) • “Randomized generalization” of NP • Can think of a game between all-knowing but potentially dishonest Merlin trying to prove statement to efficient randomized classical computer (Arthur) • If statement is true, there exists a polynomial length classical bitstring or “witness” to convince Arthur to accept with high probability (Completeness) • If statement is false, then every “witness” is rejected by Arthur with high probability (Soundness) • Under commonly believed derandomization hypothesis MA=NP 5 I.3 Quantum Merlin-Arthur • QMA: Same setup, now Arthur is BQP machine, |ψ⟩ witness is polynomial qubit quantum state • Formally: QMAm is the class of promise problems L=(Lyes,Lno) so that: • There exists a uniform verifier {�#}#∈{&,(}* of polynomial size that acts on O(m(|x|)+k(|x|)) qubits (for k∈poly(n)): k k x Lyes 0 Vx† 1 1 outVx 0 2/3 2 )9| i h | ⌦h k | | ih | | i⌦| k i ≥ x L 0 V † 1 1 V 0 1/3 2 no )8| i h | ⌦h | x | ih |out x | i⌦| i • “Quantum analogue” of NP • k-Local Hamiltonian problem is QMA-complete (when k≥2) [Kitaev ’02] / • Input: � = ∑.0( �., each term �.
    [Show full text]
  • Pseudorandom Quantum States
    Pseudorandom Quantum States Zhengfeng Ji1, Yi-Kai Liu2, and Fang Song3 1 Centre for Quantum Software and Information, School of Software, Faculty of Engineering and Information Technology, University of Technology Sydney, NSW, Australia [email protected] 2 Joint Center for Quantum Information and Computer Science (QuICS), National Institute of Standards and Technology (NIST), Gaithersburg, MD, USA, and University of Maryland, College Park, MD, USA [email protected] 3 Computer Science Department, Portland State University, Portland, OR, USA [email protected] Abstract. We propose the concept of pseudorandom quantum states, which appear random to any quantum polynomial-time adversary. It offers a computational approximation to perfectly random quantum states anal- ogous in spirit to cryptographic pseudorandom generators, as opposed to statistical notions of quantum pseudorandomness that have been studied previously, such as quantum t-designs analogous to t-wise independent distributions. Under the assumption that quantum-secure one-way functions exist, we present efficient constructions of pseudorandom states, showing that our definition is achievable. We then prove several basic properties of pseudorandom states, which show the utility of our definition. First, we show a cryptographic no-cloning theorem: no efficient quantum al- gorithm can create additional copies of a pseudorandom state, when given polynomially-many copies as input. Second, as expected for ran- dom quantum states, we show that pseudorandom quantum states are highly entangled on average. Finally, as a main application, we prove that any family of pseudorandom states naturally gives rise to a private-key quantum money scheme. 1 Introduction Pseudorandomness is a foundational concept in modern cryptography and theo- retical computer science.
    [Show full text]
  • Quantum Machine Learning
    Chapter 10 Quantum machine learning Quantum computing and machine learning are arguably two of the “hottest” topics in science at the moment. Here in Sweden, this is reflected in the fact that the two largest programs supported by the Knut and Alice Wallenberg foundation are centered around quantum technologies and artificial intelligence. In this chapter, we will discuss efforts to combine the two fields into quantum machine learning. Since this is a course about quantum algorithms, we do not cover applications of classical machine learning to simulating and understanding quantum systems, but focus instead on how machine learning can be enhance by quantum computation. We begin with a brief overview of classical machine learning and then study some examples of quantum machine learning algorithms. 10.1 A brief overview of classical machine learning What is machine learning? With the success of, and hype around, machine learning in recent years, there are examples of companies and researchers calling many things “machine learning” that would have been called something else a few years ago. According to Wikipedia, “Machine learning is the scientific study of algorithms and statistical models that computer systems use to perform a specific task without using explicit instructions, relying on patterns and inference instead”. We like the following definition (source unknown): “Machine learning studies algorithms whose performance improves with data (‘learning from experience’)”. 10.1.1 Types of machine learning Broadly speaking, there are three paradigms in machine learning for extracting meaning from some data: Unsupervised learning: Learning structure in P (data) given samples from P (data). Here, the • machine learning is used to generate knowledge by analyzing unlabelled data.
    [Show full text]
  • Quantum Cryptography
    QFFF DISSERTATION IMPERIAL COLLEGE LONDON DEPARTMENT OF PHYSICS Quantum Cryptography Author: Supervisor: Yoann PIÉTRI Jonathan HALLIWELL September 25, 2020 Submitted in partial fulfillment of the requirements for the degree of Master of Science of Imperial College London. Abstract In this paper, we review how the laws of Quantum Mechanics allow creating unconditionally secure protocols in cryptography, i.e. protocols where the security is ensured by physical laws and bounds and not on some computationally hard problems. We first review the use of Quantum Mechanics to perform unconditionally secure secret key distribution, and we then extend the ideas to other cryptographic tasks including public-key cryptography, digital signatures, and fingerprinting. Acknowledgements I would like to thanks everyone in the Theoretical Physics group for this year. I have learned a lot and it was a good ending after two years of engineering school. I thank Jonathan HALLIWELL for supervising this dissertation. I also would like to thank my family for their endless support and their help which allowed me to study in the United Kingdom this year. And finally, special thanks to Nada KAFIA, without whom I wouldn’t have gotten over these difficult times. Contents Abstract 2 Acknowledgements 3 1 Introduction 6 1.1 The beginnings of cryptography.............................6 1.2 Modern Cryptography..................................7 1.3 Quantum computer or the end of cryptography?....................9 1.4 Post-Quantum Cryptography............................... 10 2 Preliminaries 12 2.1 Notations......................................... 12 2.2 Quantum gates...................................... 13 2.3 Measurement....................................... 15 2.4 No-cloning theorem................................... 15 2.5 Conjugate coding.................................... 16 2.6 Entanglement, EPR paradox and BELL inequalities..................
    [Show full text]
  • Advances in Quantum Cryptography
    Advances in Quantum Cryptography S. Pirandola1,2, U. L. Andersen3, L. Banchi4, M. Berta5, D. Bunandar2, R. Colbeck6, D. Englund2, T. Gehring3, C. Lupo7, C. Ottaviani1, J. Pereira1, M. Razavi8, J. S. Shaari9,10, M. Tomamichel11, V. C. Usenko12, G. Vallone13, P. Villoresi13, P. Wallden14 1Computer Science and York Centre for Quantum Technologies, University of York, York YO10 5GH, UK 2Research Laboratory of Electronics, Massachusetts Institute of Technology (MIT), Cambridge, Massachusetts 02139, USA 3Center for Macroscopic Quantum States (bigQ), Department of Physics, Technical University of Denmark, Fysikvej, 2800 Kgs. Lyngby, Denmark 4Department of Physics and Astronomy, University of Florence, via G. Sansone 1, I-50019 Sesto Fiorentino (FI), Italy 5Department of Computing, Imperial College, Kensington, London SW7 2AZ, UK 6Department of Mathematics, University of York, York YO10 5DD, UK 7Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, UK 8School of Electronic and Electrical Engineering, University of Leeds, Leeds, LS2 9JT, UK 9Faculty of Science, International Islamic University Malaysia (IIUM), Jalan Sultan Ahmad Shah, 25200 Kuantan, Pahang, Malaysia 10Institute of Mathematical Research (INSPEM), University Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia 11Centre for Quantum Software and Information, School of Software, University of Technology Sydney, Sydney NSW 2007, Australia 12Department of Optics, Palacky University, 17. listopadu 50, 772 07 Olomouc, Czech Republic 13Dipartimento di Ingegneria dell’Informazione, Universit´adegli Studi di Padova, via Gradenigo 6B, 35131 Padova, Italy and 14School of Informatics, University of Edinburgh, 10 Crichton Street, Edinburgh EH8 9AB, UK Quantum cryptography is arguably the fastest growing area in quantum information science. Novel theoretical protocols are designed on a regular basis, security proofs are constantly improv- ing, and experiments are gradually moving from proof-of-principle lab demonstrations to in-field implementations and technological prototypes.
    [Show full text]